LMFDB - Embedded newform 2352.4.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +7.41421 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +7.41421 q^{5} +9.00000 q^{9} -10.4853 q^{11} +2.78680 q^{13} -22.2426 q^{15} +50.4437 q^{17} -125.054 q^{19} +182.250 q^{23} -70.0294 q^{25} -27.0000 q^{27} +156.132 q^{29} -139.632 q^{31} +31.4558 q^{33} -394.558 q^{37} -8.36039 q^{39} +197.605 q^{41} -343.294 q^{43} +66.7279 q^{45} +610.004 q^{47} -151.331 q^{51} -137.529 q^{53} -77.7401 q^{55} +375.161 q^{57} -589.436 q^{59} +247.217 q^{61} +20.6619 q^{65} +395.647 q^{67} -546.749 q^{69} -285.661 q^{71} -997.457 q^{73} +210.088 q^{75} +848.264 q^{79} +81.0000 q^{81} +210.863 q^{83} +374.000 q^{85} -468.396 q^{87} -553.487 q^{89} +418.897 q^{93} -927.176 q^{95} -903.910 q^{97} -94.3675 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9} - 4 q^{11} + 48 q^{13} - 36 q^{15} + 132 q^{17} - 120 q^{19} + 76 q^{23} - 174 q^{25} - 54 q^{27} - 112 q^{29} - 432 q^{31} + 12 q^{33} - 280 q^{37} - 144 q^{39} + 36 q^{41} + 128 q^{43} + 108 q^{45} + 264 q^{47} - 396 q^{51} + 268 q^{53} - 48 q^{55} + 360 q^{57} - 336 q^{59} - 504 q^{61} + 228 q^{65} + 384 q^{67} - 228 q^{69} + 396 q^{71} - 312 q^{73} + 522 q^{75} + 848 q^{79} + 162 q^{81} + 648 q^{83} + 748 q^{85} + 336 q^{87} - 612 q^{89} + 1296 q^{93} - 904 q^{95} - 2184 q^{97} - 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9 - 4 * q^11 + 48 * q^13 - 36 * q^15 + 132 * q^17 - 120 * q^19 + 76 * q^23 - 174 * q^25 - 54 * q^27 - 112 * q^29 - 432 * q^31 + 12 * q^33 - 280 * q^37 - 144 * q^39 + 36 * q^41 + 128 * q^43 + 108 * q^45 + 264 * q^47 - 396 * q^51 + 268 * q^53 - 48 * q^55 + 360 * q^57 - 336 * q^59 - 504 * q^61 + 228 * q^65 + 384 * q^67 - 228 * q^69 + 396 * q^71 - 312 * q^73 + 522 * q^75 + 848 * q^79 + 162 * q^81 + 648 * q^83 + 748 * q^85 + 336 * q^87 - 612 * q^89 + 1296 * q^93 - 904 * q^95 - 2184 * q^97 - 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	7.41421	0.663147	0.331574	−	0.943429i	$-0.392420\pi$
$5$	7.41421	0.663147	0.331574	+	0.943429i	$0.392420\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−10.4853	−0.287403	−0.143701	−	0.989621i	$-0.545900\pi$
$11$	−10.4853	−0.287403	−0.143701	+	0.989621i	$0.545900\pi$
$12$	0	0
$13$	2.78680	0.0594553	0.0297276	−	0.999558i	$-0.490536\pi$
$13$	2.78680	0.0594553	0.0297276	+	0.999558i	$0.490536\pi$
$14$	0	0
$15$	−22.2426	−0.382868
$16$	0	0
$17$	50.4437	0.719670	0.359835	−	0.933016i	$-0.382833\pi$
$17$	50.4437	0.719670	0.359835	+	0.933016i	$0.382833\pi$
$18$	0	0
$19$	−125.054	−1.50996	−0.754982	−	0.655746i	$-0.772354\pi$
$19$	−125.054	−1.50996	−0.754982	+	0.655746i	$0.772354\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	182.250	1.65225	0.826124	−	0.563488i	$-0.190541\pi$
$23$	182.250	1.65225	0.826124	+	0.563488i	$0.190541\pi$
$24$	0	0
$25$	−70.0294	−0.560235
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	156.132	0.999758	0.499879	−	0.866095i	$-0.333378\pi$
$29$	156.132	0.999758	0.499879	+	0.866095i	$0.333378\pi$
$30$	0	0
$31$	−139.632	−0.808991	−0.404496	−	0.914540i	$-0.632553\pi$
$31$	−139.632	−0.808991	−0.404496	+	0.914540i	$0.632553\pi$
$32$	0	0
$33$	31.4558	0.165932
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−394.558	−1.75311	−0.876554	−	0.481303i	$-0.840164\pi$
$37$	−394.558	−1.75311	−0.876554	+	0.481303i	$0.840164\pi$
$38$	0	0
$39$	−8.36039	−0.0343265
$40$	0	0
$41$	197.605	0.752701	0.376350	−	0.926477i	$-0.377179\pi$
$41$	197.605	0.752701	0.376350	+	0.926477i	$0.377179\pi$
$42$	0	0
$43$	−343.294	−1.21748	−0.608741	−	0.793369i	$-0.708325\pi$
$43$	−343.294	−1.21748	−0.608741	+	0.793369i	$0.708325\pi$
$44$	0	0
$45$	66.7279	0.221049
$46$	0	0
$47$	610.004	1.89315	0.946577	−	0.322477i	$-0.104516\pi$
$47$	610.004	1.89315	0.946577	+	0.322477i	$0.104516\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−151.331	−0.415501
$52$	0	0
$53$	−137.529	−0.356435	−0.178218	−	0.983991i	$-0.557033\pi$
$53$	−137.529	−0.356435	−0.178218	+	0.983991i	$0.557033\pi$
$54$	0	0
$55$	−77.7401	−0.190590
$56$	0	0
$57$	375.161	0.871778
$58$	0	0
$59$	−589.436	−1.30064	−0.650322	−	0.759659i	$-0.725366\pi$
$59$	−589.436	−1.30064	−0.650322	+	0.759659i	$0.725366\pi$
$60$	0	0
$61$	247.217	0.518901	0.259450	−	0.965756i	$-0.416459\pi$
$61$	247.217	0.518901	0.259450	+	0.965756i	$0.416459\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.6619	0.0394276
$66$	0	0
$67$	395.647	0.721432	0.360716	−	0.932676i	$-0.382532\pi$
$67$	395.647	0.721432	0.360716	+	0.932676i	$0.382532\pi$
$68$	0	0
$69$	−546.749	−0.953926
$70$	0	0
$71$	−285.661	−0.477489	−0.238745	−	0.971082i	$-0.576736\pi$
$71$	−285.661	−0.477489	−0.238745	+	0.971082i	$0.576736\pi$
$72$	0	0
$73$	−997.457	−1.59923	−0.799613	−	0.600515i	$-0.794962\pi$
$73$	−997.457	−1.59923	−0.799613	+	0.600515i	$0.794962\pi$
$74$	0	0
$75$	210.088	0.323452
$76$	0	0
$77$	0	0
$78$	0	0
$79$	848.264	1.20807	0.604033	−	0.796960i	$-0.293560\pi$
$79$	848.264	1.20807	0.604033	+	0.796960i	$0.293560\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	210.863	0.278858	0.139429	−	0.990232i	$-0.455473\pi$
$83$	210.863	0.278858	0.139429	+	0.990232i	$0.455473\pi$
$84$	0	0
$85$	374.000	0.477247
$86$	0	0
$87$	−468.396	−0.577211
$88$	0	0
$89$	−553.487	−0.659208	−0.329604	−	0.944119i	$-0.606915\pi$
$89$	−553.487	−0.659208	−0.329604	+	0.944119i	$0.606915\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	418.897	0.467071
$94$	0	0
$95$	−927.176	−1.00133
$96$	0	0
$97$	−903.910	−0.946166	−0.473083	−	0.881018i	$-0.656859\pi$
$97$	−903.910	−0.946166	−0.473083	+	0.881018i	$0.656859\pi$
$98$	0	0
$99$	−94.3675	−0.0958009
$100$	0	0
$101$	−313.611	−0.308965	−0.154482	−	0.987996i	$-0.549371\pi$
$101$	−313.611	−0.308965	−0.154482	+	0.987996i	$0.549371\pi$
$102$	0	0
$103$	−230.975	−0.220957	−0.110479	−	0.993878i	$-0.535238\pi$
$103$	−230.975	−0.220957	−0.110479	+	0.993878i	$0.535238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−125.486	−0.113376	−0.0566879	−	0.998392i	$-0.518054\pi$
$107$	−125.486	−0.113376	−0.0566879	+	0.998392i	$0.518054\pi$
$108$	0	0
$109$	745.527	0.655124	0.327562	−	0.944830i	$-0.393773\pi$
$109$	745.527	0.655124	0.327562	+	0.944830i	$0.393773\pi$
$110$	0	0
$111$	1183.68	1.01216
$112$	0	0
$113$	−1043.76	−0.868929	−0.434464	−	0.900689i	$-0.643062\pi$
$113$	−1043.76	−0.868929	−0.434464	+	0.900689i	$0.643062\pi$
$114$	0	0
$115$	1351.24	1.09568
$116$	0	0
$117$	25.0812	0.0198184
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1221.06	−0.917400
$122$	0	0
$123$	−592.815	−0.434572
$124$	0	0
$125$	−1445.99	−1.03467
$126$	0	0
$127$	2080.17	1.45343	0.726715	−	0.686939i	$-0.241046\pi$
$127$	2080.17	1.45343	0.726715	+	0.686939i	$0.241046\pi$
$128$	0	0
$129$	1029.88	0.702914
$130$	0	0
$131$	1269.53	0.846711	0.423355	−	0.905964i	$-0.360852\pi$
$131$	1269.53	0.846711	0.423355	+	0.905964i	$0.360852\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−200.184	−0.127623
$136$	0	0
$137$	3073.19	1.91650	0.958249	−	0.285935i	$-0.0923043\pi$
$137$	3073.19	1.91650	0.958249	+	0.285935i	$0.0923043\pi$
$138$	0	0
$139$	−1013.60	−0.618504	−0.309252	−	0.950980i	$-0.600079\pi$
$139$	−1013.60	−0.618504	−0.309252	+	0.950980i	$0.600079\pi$
$140$	0	0
$141$	−1830.01	−1.09301
$142$	0	0
$143$	−29.2203	−0.0170876
$144$	0	0
$145$	1157.60	0.662987
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1231.47	0.677087	0.338544	−	0.940951i	$-0.390066\pi$
$149$	1231.47	0.677087	0.338544	+	0.940951i	$0.390066\pi$
$150$	0	0
$151$	2244.74	1.20976	0.604881	−	0.796316i	$-0.293221\pi$
$151$	2244.74	1.20976	0.604881	+	0.796316i	$0.293221\pi$
$152$	0	0
$153$	453.993	0.239890
$154$	0	0
$155$	−1035.26	−0.536481
$156$	0	0
$157$	−3787.16	−1.92515	−0.962573	−	0.271021i	$-0.912639\pi$
$157$	−3787.16	−1.92515	−0.962573	+	0.271021i	$0.912639\pi$
$158$	0	0
$159$	412.587	0.205788
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2108.56	1.01322	0.506611	−	0.862175i	$-0.330898\pi$
$163$	2108.56	1.01322	0.506611	+	0.862175i	$0.330898\pi$
$164$	0	0
$165$	233.220	0.110037
$166$	0	0
$167$	−1502.41	−0.696170	−0.348085	−	0.937463i	$-0.613168\pi$
$167$	−1502.41	−0.696170	−0.348085	+	0.937463i	$0.613168\pi$
$168$	0	0
$169$	−2189.23	−0.996465
$170$	0	0
$171$	−1125.48	−0.503321
$172$	0	0
$173$	471.256	0.207104	0.103552	−	0.994624i	$-0.466979\pi$
$173$	471.256	0.207104	0.103552	+	0.994624i	$0.466979\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1768.31	0.750927
$178$	0	0
$179$	−1332.49	−0.556395	−0.278198	−	0.960524i	$-0.589737\pi$
$179$	−1332.49	−0.556395	−0.278198	+	0.960524i	$0.589737\pi$
$180$	0	0
$181$	−997.727	−0.409726	−0.204863	−	0.978791i	$-0.565675\pi$
$181$	−997.727	−0.409726	−0.204863	+	0.978791i	$0.565675\pi$
$182$	0	0
$183$	−741.652	−0.299587
$184$	0	0
$185$	−2925.34	−1.16257
$186$	0	0
$187$	−528.916	−0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1226.72	−0.464725	−0.232362	−	0.972629i	$-0.574646\pi$
$191$	−1226.72	−0.464725	−0.232362	+	0.972629i	$0.574646\pi$
$192$	0	0
$193$	−3479.29	−1.29764	−0.648821	−	0.760941i	$-0.724738\pi$
$193$	−3479.29	−1.29764	−0.648821	+	0.760941i	$0.724738\pi$
$194$	0	0
$195$	−61.9857	−0.0227635
$196$	0	0
$197$	3193.47	1.15495	0.577476	−	0.816408i	$-0.304038\pi$
$197$	3193.47	1.15495	0.577476	+	0.816408i	$0.304038\pi$
$198$	0	0
$199$	−1065.15	−0.379429	−0.189715	−	0.981839i	$-0.560756\pi$
$199$	−1065.15	−0.379429	−0.189715	+	0.981839i	$0.560756\pi$
$200$	0	0
$201$	−1186.94	−0.416519
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1465.09	0.499152
$206$	0	0
$207$	1640.25	0.550749
$208$	0	0
$209$	1311.22	0.433968
$210$	0	0
$211$	−2057.50	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$211$	−2057.50	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$212$	0	0
$213$	856.983	0.275678
$214$	0	0
$215$	−2545.25	−0.807371
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2992.37	0.923314
$220$	0	0
$221$	140.576	0.0427881
$222$	0	0
$223$	−2028.27	−0.609071	−0.304536	−	0.952501i	$-0.598501\pi$
$223$	−2028.27	−0.609071	−0.304536	+	0.952501i	$0.598501\pi$
$224$	0	0
$225$	−630.265	−0.186745
$226$	0	0
$227$	−2423.96	−0.708739	−0.354369	−	0.935106i	$-0.615304\pi$
$227$	−2423.96	−0.708739	−0.354369	+	0.935106i	$0.615304\pi$
$228$	0	0
$229$	−1967.17	−0.567660	−0.283830	−	0.958875i	$-0.591605\pi$
$229$	−1967.17	−0.567660	−0.283830	+	0.958875i	$0.591605\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4478.33	1.25916	0.629582	−	0.776934i	$-0.283226\pi$
$233$	4478.33	1.25916	0.629582	+	0.776934i	$0.283226\pi$
$234$	0	0
$235$	4522.70	1.25544
$236$	0	0
$237$	−2544.79	−0.697477
$238$	0	0
$239$	−6116.92	−1.65553	−0.827763	−	0.561078i	$-0.810387\pi$
$239$	−6116.92	−1.65553	−0.827763	+	0.561078i	$0.810387\pi$
$240$	0	0
$241$	6228.38	1.66475	0.832376	−	0.554211i	$-0.186980\pi$
$241$	6228.38	1.66475	0.832376	+	0.554211i	$0.186980\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−348.500	−0.0897753
$248$	0	0
$249$	−632.589	−0.160999
$250$	0	0
$251$	−5904.42	−1.48479	−0.742397	−	0.669960i	$-0.766311\pi$
$251$	−5904.42	−1.48479	−0.742397	+	0.669960i	$0.766311\pi$
$252$	0	0
$253$	−1910.94	−0.474861
$254$	0	0
$255$	−1122.00	−0.275539
$256$	0	0
$257$	−408.223	−0.0990827	−0.0495414	−	0.998772i	$-0.515776\pi$
$257$	−408.223	−0.0990827	−0.0495414	+	0.998772i	$0.515776\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1405.19	0.333253
$262$	0	0
$263$	4626.01	1.08461	0.542304	−	0.840182i	$-0.317552\pi$
$263$	4626.01	1.08461	0.542304	+	0.840182i	$0.317552\pi$
$264$	0	0
$265$	−1019.67	−0.236369
$266$	0	0
$267$	1660.46	0.380594
$268$	0	0
$269$	−871.293	−0.197486	−0.0987429	−	0.995113i	$-0.531482\pi$
$269$	−871.293	−0.197486	−0.0987429	+	0.995113i	$0.531482\pi$
$270$	0	0
$271$	−6472.35	−1.45080	−0.725401	−	0.688327i	$-0.758345\pi$
$271$	−6472.35	−1.45080	−0.725401	+	0.688327i	$0.758345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	734.278	0.161013
$276$	0	0
$277$	4711.88	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$277$	4711.88	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$278$	0	0
$279$	−1256.69	−0.269664
$280$	0	0
$281$	−7165.66	−1.52124	−0.760618	−	0.649200i	$-0.775104\pi$
$281$	−7165.66	−1.52124	−0.760618	+	0.649200i	$0.775104\pi$
$282$	0	0
$283$	−6347.00	−1.33318	−0.666590	−	0.745424i	$-0.732247\pi$
$283$	−6347.00	−1.33318	−0.666590	+	0.745424i	$0.732247\pi$
$284$	0	0
$285$	2781.53	0.578117
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2368.44	−0.482076
$290$	0	0
$291$	2711.73	0.546269
$292$	0	0
$293$	9233.78	1.84110	0.920552	−	0.390621i	$-0.127740\pi$
$293$	9233.78	1.84110	0.920552	+	0.390621i	$0.127740\pi$
$294$	0	0
$295$	−4370.20	−0.862519
$296$	0	0
$297$	283.103	0.0553107
$298$	0	0
$299$	507.893	0.0982348
$300$	0	0
$301$	0	0
$302$	0	0
$303$	940.833	0.178381
$304$	0	0
$305$	1832.92	0.344108
$306$	0	0
$307$	−6786.53	−1.26165	−0.630827	−	0.775923i	$-0.717284\pi$
$307$	−6786.53	−1.26165	−0.630827	+	0.775923i	$0.717284\pi$
$308$	0	0
$309$	692.924	0.127570
$310$	0	0
$311$	−5136.77	−0.936590	−0.468295	−	0.883572i	$-0.655132\pi$
$311$	−5136.77	−0.936590	−0.468295	+	0.883572i	$0.655132\pi$
$312$	0	0
$313$	−3763.56	−0.679645	−0.339822	−	0.940490i	$-0.610367\pi$
$313$	−3763.56	−0.679645	−0.339822	+	0.940490i	$0.610367\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1034.95	−0.183370	−0.0916851	−	0.995788i	$-0.529225\pi$
$317$	−1034.95	−0.183370	−0.0916851	+	0.995788i	$0.529225\pi$
$318$	0	0
$319$	−1637.09	−0.287333
$320$	0	0
$321$	376.458	0.0654575
$322$	0	0
$323$	−6308.17	−1.08668
$324$	0	0
$325$	−195.158	−0.0333089
$326$	0	0
$327$	−2236.58	−0.378236
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8800.06	−1.46131	−0.730657	−	0.682744i	$-0.760786\pi$
$331$	−8800.06	−1.46131	−0.730657	+	0.682744i	$0.760786\pi$
$332$	0	0
$333$	−3551.03	−0.584369
$334$	0	0
$335$	2933.41	0.478416
$336$	0	0
$337$	−5859.78	−0.947189	−0.473595	−	0.880743i	$-0.657044\pi$
$337$	−5859.78	−0.947189	−0.473595	+	0.880743i	$0.657044\pi$
$338$	0	0
$339$	3131.29	0.501676
$340$	0	0
$341$	1464.09	0.232506
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4053.72	−0.632593
$346$	0	0
$347$	−7938.54	−1.22814	−0.614068	−	0.789253i	$-0.710468\pi$
$347$	−7938.54	−1.22814	−0.614068	+	0.789253i	$0.710468\pi$
$348$	0	0
$349$	−9927.75	−1.52269	−0.761347	−	0.648344i	$-0.775462\pi$
$349$	−9927.75	−1.52269	−0.761347	+	0.648344i	$0.775462\pi$
$350$	0	0
$351$	−75.2435	−0.0114422
$352$	0	0
$353$	−10103.0	−1.52332	−0.761658	−	0.647979i	$-0.775614\pi$
$353$	−10103.0	−1.52332	−0.761658	+	0.647979i	$0.775614\pi$
$354$	0	0
$355$	−2117.95	−0.316646
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4825.27	0.709382	0.354691	−	0.934984i	$-0.384586\pi$
$359$	4825.27	0.709382	0.354691	+	0.934984i	$0.384586\pi$
$360$	0	0
$361$	8779.46	1.27999
$362$	0	0
$363$	3663.18	0.529661
$364$	0	0
$365$	−7395.36	−1.06052
$366$	0	0
$367$	−6935.30	−0.986430	−0.493215	−	0.869907i	$-0.664178\pi$
$367$	−6935.30	−0.986430	−0.493215	+	0.869907i	$0.664178\pi$
$368$	0	0
$369$	1778.45	0.250900
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14093.1	1.95634	0.978168	−	0.207816i	$-0.0666355\pi$
$373$	14093.1	1.95634	0.978168	+	0.207816i	$0.0666355\pi$
$374$	0	0
$375$	4337.97	0.597365
$376$	0	0
$377$	435.108	0.0594409
$378$	0	0
$379$	−5354.17	−0.725661	−0.362830	−	0.931855i	$-0.618190\pi$
$379$	−5354.17	−0.725661	−0.362830	+	0.931855i	$0.618190\pi$
$380$	0	0
$381$	−6240.52	−0.839138
$382$	0	0
$383$	8970.47	1.19679	0.598394	−	0.801202i	$-0.295806\pi$
$383$	8970.47	1.19679	0.598394	+	0.801202i	$0.295806\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3089.64	−0.405828
$388$	0	0
$389$	−3702.58	−0.482592	−0.241296	−	0.970452i	$-0.577572\pi$
$389$	−3702.58	−0.482592	−0.241296	+	0.970452i	$0.577572\pi$
$390$	0	0
$391$	9193.34	1.18907
$392$	0	0
$393$	−3808.58	−0.488849
$394$	0	0
$395$	6289.21	0.801125
$396$	0	0
$397$	−2083.24	−0.263362	−0.131681	−	0.991292i	$-0.542037\pi$
$397$	−2083.24	−0.263362	−0.131681	+	0.991292i	$0.542037\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10634.0	−1.32428	−0.662141	−	0.749379i	$-0.730352\pi$
$401$	−10634.0	−1.32428	−0.662141	+	0.749379i	$0.730352\pi$
$402$	0	0
$403$	−389.127	−0.0480988
$404$	0	0
$405$	600.551	0.0736830
$406$	0	0
$407$	4137.06	0.503848
$408$	0	0
$409$	−6516.37	−0.787808	−0.393904	−	0.919152i	$-0.628876\pi$
$409$	−6516.37	−0.787808	−0.393904	+	0.919152i	$0.628876\pi$
$410$	0	0
$411$	−9219.56	−1.10649
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1563.38	0.184924
$416$	0	0
$417$	3040.79	0.357094
$418$	0	0
$419$	−6079.92	−0.708887	−0.354443	−	0.935077i	$-0.615330\pi$
$419$	−6079.92	−0.708887	−0.354443	+	0.935077i	$0.615330\pi$
$420$	0	0
$421$	−5631.58	−0.651939	−0.325969	−	0.945380i	$-0.605691\pi$
$421$	−5631.58	−0.651939	−0.325969	+	0.945380i	$0.605691\pi$
$422$	0	0
$423$	5490.04	0.631051
$424$	0	0
$425$	−3532.54	−0.403184
$426$	0	0
$427$	0	0
$428$	0	0
$429$	87.6610	0.00986554
$430$	0	0
$431$	3736.90	0.417633	0.208817	−	0.977955i	$-0.433039\pi$
$431$	3736.90	0.417633	0.208817	+	0.977955i	$0.433039\pi$
$432$	0	0
$433$	−5757.46	−0.638998	−0.319499	−	0.947587i	$-0.603515\pi$
$433$	−5757.46	−0.638998	−0.319499	+	0.947587i	$0.603515\pi$
$434$	0	0
$435$	−3472.79	−0.382776
$436$	0	0
$437$	−22791.0	−2.49484
$438$	0	0
$439$	9812.76	1.06683	0.533414	−	0.845854i	$-0.320909\pi$
$439$	9812.76	1.06683	0.533414	+	0.845854i	$0.320909\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5830.30	−0.625296	−0.312648	−	0.949869i	$-0.601216\pi$
$443$	−5830.30	−0.625296	−0.312648	+	0.949869i	$0.601216\pi$
$444$	0	0
$445$	−4103.67	−0.437152
$446$	0	0
$447$	−3694.41	−0.390916
$448$	0	0
$449$	−8674.94	−0.911794	−0.455897	−	0.890033i	$-0.650681\pi$
$449$	−8674.94	−0.911794	−0.455897	+	0.890033i	$0.650681\pi$
$450$	0	0
$451$	−2071.95	−0.216328
$452$	0	0
$453$	−6734.21	−0.698456
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9106.82	0.932165	0.466082	−	0.884741i	$-0.345665\pi$
$457$	9106.82	0.932165	0.466082	+	0.884741i	$0.345665\pi$
$458$	0	0
$459$	−1361.98	−0.138500
$460$	0	0
$461$	−8729.69	−0.881957	−0.440979	−	0.897518i	$-0.645369\pi$
$461$	−8729.69	−0.881957	−0.440979	+	0.897518i	$0.645369\pi$
$462$	0	0
$463$	1795.62	0.180237	0.0901184	−	0.995931i	$-0.471275\pi$
$463$	1795.62	0.180237	0.0901184	+	0.995931i	$0.471275\pi$
$464$	0	0
$465$	3105.79	0.309737
$466$	0	0
$467$	−130.559	−0.0129370	−0.00646849	−	0.999979i	$-0.502059\pi$
$467$	−130.559	−0.0129370	−0.00646849	+	0.999979i	$0.502059\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	11361.5	1.11148
$472$	0	0
$473$	3599.53	0.349908
$474$	0	0
$475$	8757.45	0.845935
$476$	0	0
$477$	−1237.76	−0.118812
$478$	0	0
$479$	−11845.8	−1.12996	−0.564978	−	0.825106i	$-0.691115\pi$
$479$	−11845.8	−1.12996	−0.564978	+	0.825106i	$0.691115\pi$
$480$	0	0
$481$	−1099.55	−0.104232
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6701.78	−0.627448
$486$	0	0
$487$	4807.21	0.447301	0.223650	−	0.974669i	$-0.428203\pi$
$487$	4807.21	0.447301	0.223650	+	0.974669i	$0.428203\pi$
$488$	0	0
$489$	−6325.68	−0.584984
$490$	0	0
$491$	−6068.04	−0.557733	−0.278866	−	0.960330i	$-0.589959\pi$
$491$	−6068.04	−0.557733	−0.278866	+	0.960330i	$0.589959\pi$
$492$	0	0
$493$	7875.87	0.719496
$494$	0	0
$495$	−699.661	−0.0635302
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15506.4	1.39111	0.695554	−	0.718474i	$-0.255159\pi$
$499$	15506.4	1.39111	0.695554	+	0.718474i	$0.255159\pi$
$500$	0	0
$501$	4507.24	0.401934
$502$	0	0
$503$	1496.79	0.132681	0.0663405	−	0.997797i	$-0.478868\pi$
$503$	1496.79	0.132681	0.0663405	+	0.997797i	$0.478868\pi$
$504$	0	0
$505$	−2325.18	−0.204889
$506$	0	0
$507$	6567.70	0.575309
$508$	0	0
$509$	−5053.74	−0.440084	−0.220042	−	0.975490i	$-0.570620\pi$
$509$	−5053.74	−0.440084	−0.220042	+	0.975490i	$0.570620\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3376.45	0.290593
$514$	0	0
$515$	−1712.50	−0.146527
$516$	0	0
$517$	−6396.07	−0.544098
$518$	0	0
$519$	−1413.77	−0.119571
$520$	0	0
$521$	9736.73	0.818760	0.409380	−	0.912364i	$-0.365745\pi$
$521$	9736.73	0.818760	0.409380	+	0.912364i	$0.365745\pi$
$522$	0	0
$523$	11796.7	0.986295	0.493148	−	0.869946i	$-0.335846\pi$
$523$	11796.7	0.986295	0.493148	+	0.869946i	$0.335846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7043.57	−0.582206
$528$	0	0
$529$	21048.0	1.72992
$530$	0	0
$531$	−5304.92	−0.433548
$532$	0	0
$533$	550.685	0.0447520
$534$	0	0
$535$	−930.381	−0.0751848
$536$	0	0
$537$	3997.46	0.321235
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4310.82	−0.342581	−0.171291	−	0.985221i	$-0.554794\pi$
$541$	−4310.82	−0.342581	−0.171291	+	0.985221i	$0.554794\pi$
$542$	0	0
$543$	2993.18	0.236556
$544$	0	0
$545$	5527.50	0.434444
$546$	0	0
$547$	−17015.9	−1.33007	−0.665034	−	0.746813i	$-0.731583\pi$
$547$	−17015.9	−1.33007	−0.665034	+	0.746813i	$0.731583\pi$
$548$	0	0
$549$	2224.96	0.172967
$550$	0	0
$551$	−19524.9	−1.50960
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8776.02	0.671210
$556$	0	0
$557$	1741.18	0.132452	0.0662262	−	0.997805i	$-0.478904\pi$
$557$	1741.18	0.132452	0.0662262	+	0.997805i	$0.478904\pi$
$558$	0	0
$559$	−956.689	−0.0723858
$560$	0	0
$561$	1586.75	0.119416
$562$	0	0
$563$	−11346.8	−0.849401	−0.424700	−	0.905334i	$-0.639621\pi$
$563$	−11346.8	−0.849401	−0.424700	+	0.905334i	$0.639621\pi$
$564$	0	0
$565$	−7738.68	−0.576228
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18417.4	−1.35694	−0.678470	−	0.734628i	$-0.737357\pi$
$569$	−18417.4	−1.35694	−0.678470	+	0.734628i	$0.737357\pi$
$570$	0	0
$571$	−9998.49	−0.732791	−0.366396	−	0.930459i	$-0.619408\pi$
$571$	−9998.49	−0.732791	−0.366396	+	0.930459i	$0.619408\pi$
$572$	0	0
$573$	3680.16	0.268309
$574$	0	0
$575$	−12762.8	−0.925648
$576$	0	0
$577$	−1401.71	−0.101133	−0.0505667	−	0.998721i	$-0.516103\pi$
$577$	−1401.71	−0.101133	−0.0505667	+	0.998721i	$0.516103\pi$
$578$	0	0
$579$	10437.9	0.749194
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1442.03	0.102440
$584$	0	0
$585$	185.957	0.0131425
$586$	0	0
$587$	10851.3	0.763001	0.381500	−	0.924369i	$-0.375407\pi$
$587$	10851.3	0.763001	0.381500	+	0.924369i	$0.375407\pi$
$588$	0	0
$589$	17461.6	1.22155
$590$	0	0
$591$	−9580.42	−0.666812
$592$	0	0
$593$	20462.0	1.41699	0.708493	−	0.705717i	$-0.249375\pi$
$593$	20462.0	1.41699	0.708493	+	0.705717i	$0.249375\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3195.45	0.219064
$598$	0	0
$599$	−9990.42	−0.681465	−0.340733	−	0.940160i	$-0.610675\pi$
$599$	−9990.42	−0.681465	−0.340733	+	0.940160i	$0.610675\pi$
$600$	0	0
$601$	17435.9	1.18341	0.591703	−	0.806156i	$-0.298456\pi$
$601$	17435.9	1.18341	0.591703	+	0.806156i	$0.298456\pi$
$602$	0	0
$603$	3560.82	0.240477
$604$	0	0
$605$	−9053.19	−0.608371
$606$	0	0
$607$	−16700.3	−1.11671	−0.558356	−	0.829601i	$-0.688568\pi$
$607$	−16700.3	−1.11671	−0.558356	+	0.829601i	$0.688568\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1699.96	0.112558
$612$	0	0
$613$	−26702.9	−1.75941	−0.879707	−	0.475515i	$-0.842262\pi$
$613$	−26702.9	−1.75941	−0.879707	+	0.475515i	$0.842262\pi$
$614$	0	0
$615$	−4395.26	−0.288185
$616$	0	0
$617$	27790.4	1.81329	0.906645	−	0.421894i	$-0.138635\pi$
$617$	27790.4	1.81329	0.906645	+	0.421894i	$0.138635\pi$
$618$	0	0
$619$	−1736.08	−0.112728	−0.0563642	−	0.998410i	$-0.517951\pi$
$619$	−1736.08	−0.112728	−0.0563642	+	0.998410i	$0.517951\pi$
$620$	0	0
$621$	−4920.74	−0.317975
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1967.20	−0.125901
$626$	0	0
$627$	−3933.67	−0.250552
$628$	0	0
$629$	−19903.0	−1.26166
$630$	0	0
$631$	8990.27	0.567190	0.283595	−	0.958944i	$-0.408473\pi$
$631$	8990.27	0.567190	0.283595	+	0.958944i	$0.408473\pi$
$632$	0	0
$633$	6172.49	0.387574
$634$	0	0
$635$	15422.9	0.963839
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2570.95	−0.159163
$640$	0	0
$641$	−13769.6	−0.848467	−0.424233	−	0.905553i	$-0.639456\pi$
$641$	−13769.6	−0.848467	−0.424233	+	0.905553i	$0.639456\pi$
$642$	0	0
$643$	−26969.9	−1.65411	−0.827053	−	0.562124i	$-0.809985\pi$
$643$	−26969.9	−1.65411	−0.827053	+	0.562124i	$0.809985\pi$
$644$	0	0
$645$	7635.75	0.466136
$646$	0	0
$647$	−24401.7	−1.48274	−0.741368	−	0.671098i	$-0.765823\pi$
$647$	−24401.7	−1.48274	−0.741368	+	0.671098i	$0.765823\pi$
$648$	0	0
$649$	6180.40	0.373809
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15969.6	0.957026	0.478513	−	0.878080i	$-0.341176\pi$
$653$	15969.6	0.957026	0.478513	+	0.878080i	$0.341176\pi$
$654$	0	0
$655$	9412.55	0.561494
$656$	0	0
$657$	−8977.11	−0.533075
$658$	0	0
$659$	11596.2	0.685467	0.342733	−	0.939433i	$-0.388647\pi$
$659$	11596.2	0.685467	0.342733	+	0.939433i	$0.388647\pi$
$660$	0	0
$661$	12602.2	0.741558	0.370779	−	0.928721i	$-0.379091\pi$
$661$	12602.2	0.741558	0.370779	+	0.928721i	$0.379091\pi$
$662$	0	0
$663$	−421.729	−0.0247037
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28455.0	1.65185
$668$	0	0
$669$	6084.80	0.351647
$670$	0	0
$671$	−2592.14	−0.149134
$672$	0	0
$673$	2126.29	0.121787	0.0608934	−	0.998144i	$-0.480605\pi$
$673$	2126.29	0.121787	0.0608934	+	0.998144i	$0.480605\pi$
$674$	0	0
$675$	1890.79	0.107817
$676$	0	0
$677$	2619.38	0.148702	0.0743508	−	0.997232i	$-0.476312\pi$
$677$	2619.38	0.148702	0.0743508	+	0.997232i	$0.476312\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7271.87	0.409190
$682$	0	0
$683$	29929.8	1.67677	0.838383	−	0.545082i	$-0.183501\pi$
$683$	29929.8	1.67677	0.838383	+	0.545082i	$0.183501\pi$
$684$	0	0
$685$	22785.3	1.27092
$686$	0	0
$687$	5901.51	0.327739
$688$	0	0
$689$	−383.265	−0.0211919
$690$	0	0
$691$	−6760.90	−0.372209	−0.186105	−	0.982530i	$-0.559586\pi$
$691$	−6760.90	−0.372209	−0.186105	+	0.982530i	$0.559586\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7515.02	−0.410160
$696$	0	0
$697$	9967.92	0.541696
$698$	0	0
$699$	−13435.0	−0.726979
$700$	0	0
$701$	467.205	0.0251727	0.0125864	−	0.999921i	$-0.495994\pi$
$701$	467.205	0.0251727	0.0125864	+	0.999921i	$0.495994\pi$
$702$	0	0
$703$	49341.0	2.64713
$704$	0	0
$705$	−13568.1	−0.724829
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8824.64	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$709$	8824.64	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$710$	0	0
$711$	7634.38	0.402688
$712$	0	0
$713$	−25448.0	−1.33665
$714$	0	0
$715$	−216.646	−0.0113316
$716$	0	0
$717$	18350.8	0.955818
$718$	0	0
$719$	21098.9	1.09438	0.547188	−	0.837010i	$-0.315698\pi$
$719$	21098.9	1.09438	0.547188	+	0.837010i	$0.315698\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18685.1	−0.961145
$724$	0	0
$725$	−10933.8	−0.560100
$726$	0	0
$727$	4616.48	0.235510	0.117755	−	0.993043i	$-0.462430\pi$
$727$	4616.48	0.235510	0.117755	+	0.993043i	$0.462430\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17317.0	−0.876185
$732$	0	0
$733$	−10688.1	−0.538571	−0.269285	−	0.963060i	$-0.586788\pi$
$733$	−10688.1	−0.538571	−0.269285	+	0.963060i	$0.586788\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4148.47	−0.207342
$738$	0	0
$739$	15367.6	0.764959	0.382480	−	0.923964i	$-0.375070\pi$
$739$	15367.6	0.764959	0.382480	+	0.923964i	$0.375070\pi$
$740$	0	0
$741$	1045.50	0.0518318
$742$	0	0
$743$	−6502.58	−0.321072	−0.160536	−	0.987030i	$-0.551322\pi$
$743$	−6502.58	−0.321072	−0.160536	+	0.987030i	$0.551322\pi$
$744$	0	0
$745$	9130.38	0.449009
$746$	0	0
$747$	1897.77	0.0929527
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19874.1	0.965670	0.482835	−	0.875711i	$-0.339607\pi$
$751$	19874.1	0.965670	0.482835	+	0.875711i	$0.339607\pi$
$752$	0	0
$753$	17713.2	0.857247
$754$	0	0
$755$	16642.9	0.802250
$756$	0	0
$757$	−15157.8	−0.727765	−0.363883	−	0.931445i	$-0.618549\pi$
$757$	−15157.8	−0.727765	−0.363883	+	0.931445i	$0.618549\pi$
$758$	0	0
$759$	5732.82	0.274161
$760$	0	0
$761$	35219.8	1.67768	0.838842	−	0.544375i	$-0.183233\pi$
$761$	35219.8	1.67768	0.838842	+	0.544375i	$0.183233\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3366.00	0.159082
$766$	0	0
$767$	−1642.64	−0.0773301
$768$	0	0
$769$	−2264.35	−0.106183	−0.0530915	−	0.998590i	$-0.516907\pi$
$769$	−2264.35	−0.106183	−0.0530915	+	0.998590i	$0.516907\pi$
$770$	0	0
$771$	1224.67	0.0572054
$772$	0	0
$773$	−9532.79	−0.443558	−0.221779	−	0.975097i	$-0.571186\pi$
$773$	−9532.79	−0.443558	−0.221779	+	0.975097i	$0.571186\pi$
$774$	0	0
$775$	9778.38	0.453226
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24711.3	−1.13655
$780$	0	0
$781$	2995.24	0.137232
$782$	0	0
$783$	−4215.56	−0.192404
$784$	0	0
$785$	−28078.8	−1.27666
$786$	0	0
$787$	33213.0	1.50434	0.752170	−	0.658969i	$-0.229007\pi$
$787$	33213.0	1.50434	0.752170	+	0.658969i	$0.229007\pi$
$788$	0	0
$789$	−13878.0	−0.626199
$790$	0	0
$791$	0	0
$792$	0	0
$793$	688.945	0.0308514
$794$	0	0
$795$	3059.01	0.136468
$796$	0	0
$797$	42065.6	1.86956	0.934781	−	0.355223i	$-0.115595\pi$
$797$	42065.6	1.86956	0.934781	+	0.355223i	$0.115595\pi$
$798$	0	0
$799$	30770.8	1.36245
$800$	0	0
$801$	−4981.39	−0.219736
$802$	0	0
$803$	10458.6	0.459622
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2613.88	0.114018
$808$	0	0
$809$	−777.560	−0.0337918	−0.0168959	−	0.999857i	$-0.505378\pi$
$809$	−777.560	−0.0337918	−0.0168959	+	0.999857i	$0.505378\pi$
$810$	0	0
$811$	−16559.4	−0.716991	−0.358496	−	0.933531i	$-0.616710\pi$
$811$	−16559.4	−0.716991	−0.358496	+	0.933531i	$0.616710\pi$
$812$	0	0
$813$	19417.0	0.837620
$814$	0	0
$815$	15633.3	0.671916
$816$	0	0
$817$	42930.2	1.83836
$818$	0	0
$819$	0	0
$820$	0	0
$821$	231.224	0.00982921	0.00491460	−	0.999988i	$-0.498436\pi$
$821$	231.224	0.00982921	0.00491460	+	0.999988i	$0.498436\pi$
$822$	0	0
$823$	5222.86	0.221212	0.110606	−	0.993864i	$-0.464721\pi$
$823$	5222.86	0.221212	0.110606	+	0.993864i	$0.464721\pi$
$824$	0	0
$825$	−2202.84	−0.0929611
$826$	0	0
$827$	46225.5	1.94367	0.971836	−	0.235658i	$-0.0757246\pi$
$827$	46225.5	1.94367	0.971836	+	0.235658i	$0.0757246\pi$
$828$	0	0
$829$	−39594.2	−1.65882	−0.829410	−	0.558640i	$-0.811323\pi$
$829$	−39594.2	−1.65882	−0.829410	+	0.558640i	$0.811323\pi$
$830$	0	0
$831$	−14135.6	−0.590084
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11139.2	−0.461663
$836$	0	0
$837$	3770.08	0.155690
$838$	0	0
$839$	45737.6	1.88205	0.941023	−	0.338344i	$-0.109867\pi$
$839$	45737.6	1.88205	0.941023	+	0.338344i	$0.109867\pi$
$840$	0	0
$841$	−11.7878	−0.000483326 0
$842$	0	0
$843$	21497.0	0.878286
$844$	0	0
$845$	−16231.4	−0.660803
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19041.0	0.769712
$850$	0	0
$851$	−71908.2	−2.89657
$852$	0	0
$853$	−8795.55	−0.353053	−0.176526	−	0.984296i	$-0.556486\pi$
$853$	−8795.55	−0.353053	−0.176526	+	0.984296i	$0.556486\pi$
$854$	0	0
$855$	−8344.58	−0.333776
$856$	0	0
$857$	30254.9	1.20594	0.602969	−	0.797764i	$-0.293984\pi$
$857$	30254.9	1.20594	0.602969	+	0.797764i	$0.293984\pi$
$858$	0	0
$859$	−44993.5	−1.78715	−0.893573	−	0.448918i	$-0.851809\pi$
$859$	−44993.5	−1.78715	−0.893573	+	0.448918i	$0.851809\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44667.9	−1.76189	−0.880946	−	0.473217i	$-0.843093\pi$
$863$	−44667.9	−1.76189	−0.880946	+	0.473217i	$0.843093\pi$
$864$	0	0
$865$	3493.99	0.137340
$866$	0	0
$867$	7105.31	0.278327
$868$	0	0
$869$	−8894.29	−0.347201
$870$	0	0
$871$	1102.59	0.0428929
$872$	0	0
$873$	−8135.19	−0.315389
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5986.84	0.230515	0.115257	−	0.993336i	$-0.463231\pi$
$877$	5986.84	0.230515	0.115257	+	0.993336i	$0.463231\pi$
$878$	0	0
$879$	−27701.3	−1.06296
$880$	0	0
$881$	37911.8	1.44981	0.724904	−	0.688850i	$-0.241884\pi$
$881$	37911.8	1.44981	0.724904	+	0.688850i	$0.241884\pi$
$882$	0	0
$883$	16293.6	0.620978	0.310489	−	0.950577i	$-0.399507\pi$
$883$	16293.6	0.620978	0.310489	+	0.950577i	$0.399507\pi$
$884$	0	0
$885$	13110.6	0.497975
$886$	0	0
$887$	−8853.41	−0.335139	−0.167570	−	0.985860i	$-0.553592\pi$
$887$	−8853.41	−0.335139	−0.167570	+	0.985860i	$0.553592\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−849.308	−0.0319336
$892$	0	0
$893$	−76283.4	−2.85859
$894$	0	0
$895$	−9879.34	−0.368972
$896$	0	0
$897$	−1523.68	−0.0567159
$898$	0	0
$899$	−21801.1	−0.808796
$900$	0	0
$901$	−6937.47	−0.256516
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7397.36	−0.271709
$906$	0	0
$907$	−16662.5	−0.609998	−0.304999	−	0.952353i	$-0.598656\pi$
$907$	−16662.5	−0.609998	−0.304999	+	0.952353i	$0.598656\pi$
$908$	0	0
$909$	−2822.50	−0.102988
$910$	0	0
$911$	29071.2	1.05727	0.528635	−	0.848849i	$-0.322704\pi$
$911$	29071.2	1.05727	0.528635	+	0.848849i	$0.322704\pi$
$912$	0	0
$913$	−2210.96	−0.0801446
$914$	0	0
$915$	−5498.77	−0.198671
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−7973.39	−0.286200	−0.143100	−	0.989708i	$-0.545707\pi$
$919$	−7973.39	−0.286200	−0.143100	+	0.989708i	$0.545707\pi$
$920$	0	0
$921$	20359.6	0.728416
$922$	0	0
$923$	−796.079	−0.0283892
$924$	0	0
$925$	27630.7	0.982154
$926$	0	0
$927$	−2078.77	−0.0736525
$928$	0	0
$929$	30141.5	1.06449	0.532244	−	0.846591i	$-0.321349\pi$
$929$	30141.5	1.06449	0.532244	+	0.846591i	$0.321349\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15410.3	0.540740
$934$	0	0
$935$	−3921.50	−0.137162
$936$	0	0
$937$	1126.37	0.0392709	0.0196354	−	0.999807i	$-0.493749\pi$
$937$	1126.37	0.0392709	0.0196354	+	0.999807i	$0.493749\pi$
$938$	0	0
$939$	11290.7	0.392393
$940$	0	0
$941$	−10333.3	−0.357976	−0.178988	−	0.983851i	$-0.557282\pi$
$941$	−10333.3	−0.357976	−0.178988	+	0.983851i	$0.557282\pi$
$942$	0	0
$943$	36013.5	1.24365
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24810.3	−0.851347	−0.425673	−	0.904877i	$-0.639963\pi$
$947$	−24810.3	−0.851347	−0.425673	+	0.904877i	$0.639963\pi$
$948$	0	0
$949$	−2779.71	−0.0950824
$950$	0	0
$951$	3104.84	0.105869
$952$	0	0
$953$	−15048.6	−0.511513	−0.255757	−	0.966741i	$-0.582325\pi$
$953$	−15048.6	−0.511513	−0.255757	+	0.966741i	$0.582325\pi$
$954$	0	0
$955$	−9095.17	−0.308181
$956$	0	0
$957$	4911.26	0.165892
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10293.8	−0.345533
$962$	0	0
$963$	−1129.38	−0.0377919
$964$	0	0
$965$	−25796.2	−0.860528
$966$	0	0
$967$	15619.9	0.519442	0.259721	−	0.965684i	$-0.416369\pi$
$967$	15619.9	0.519442	0.259721	+	0.965684i	$0.416369\pi$
$968$	0	0
$969$	18924.5	0.627392
$970$	0	0
$971$	24833.0	0.820730	0.410365	−	0.911921i	$-0.365401\pi$
$971$	24833.0	0.820730	0.410365	+	0.911921i	$0.365401\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	585.473	0.0192309
$976$	0	0
$977$	38656.8	1.26586	0.632928	−	0.774211i	$-0.281853\pi$
$977$	38656.8	1.26586	0.632928	+	0.774211i	$0.281853\pi$
$978$	0	0
$979$	5803.47	0.189458
$980$	0	0
$981$	6709.75	0.218375
$982$	0	0
$983$	−26664.1	−0.865162	−0.432581	−	0.901595i	$-0.642397\pi$
$983$	−26664.1	−0.865162	−0.432581	+	0.901595i	$0.642397\pi$
$984$	0	0
$985$	23677.1	0.765903
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62565.2	−2.01158
$990$	0	0
$991$	−27228.2	−0.872786	−0.436393	−	0.899756i	$-0.643744\pi$
$991$	−27228.2	−0.872786	−0.436393	+	0.899756i	$0.643744\pi$
$992$	0	0
$993$	26400.2	0.843690
$994$	0	0
$995$	−7897.24	−0.251617
$996$	0	0
$997$	5162.91	0.164003	0.0820016	−	0.996632i	$-0.473869\pi$
$997$	5162.91	0.164003	0.0820016	+	0.996632i	$0.473869\pi$
$998$	0	0
$999$	10653.1	0.337386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bu.1.2		2
4.3	odd	2		294.4.a.o.1.2	yes	2
7.6	odd	2		2352.4.a.bw.1.1		2
12.11	even	2		882.4.a.t.1.1		2
28.3	even	6		294.4.e.m.79.2		4
28.11	odd	6		294.4.e.k.79.1		4
28.19	even	6		294.4.e.m.67.2		4
28.23	odd	6		294.4.e.k.67.1		4
28.27	even	2		294.4.a.l.1.1	✓	2
84.11	even	6		882.4.g.bk.667.2		4
84.23	even	6		882.4.g.bk.361.2		4
84.47	odd	6		882.4.g.be.361.1		4
84.59	odd	6		882.4.g.be.667.1		4
84.83	odd	2		882.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.l.1.1	✓	2	28.27	even	2
294.4.a.o.1.2	yes	2	4.3	odd	2
294.4.e.k.67.1		4	28.23	odd	6
294.4.e.k.79.1		4	28.11	odd	6
294.4.e.m.67.2		4	28.19	even	6
294.4.e.m.79.2		4	28.3	even	6
882.4.a.t.1.1		2	12.11	even	2
882.4.a.bb.1.2		2	84.83	odd	2
882.4.g.be.361.1		4	84.47	odd	6
882.4.g.be.667.1		4	84.59	odd	6
882.4.g.bk.361.2		4	84.23	even	6
882.4.g.bk.667.2		4	84.11	even	6
2352.4.a.bu.1.2		2	1.1	even	1	trivial
2352.4.a.bw.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +7.41421 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +7.41421 q^{5} +9.00000 q^{9} -10.4853 q^{11} +2.78680 q^{13} -22.2426 q^{15} +50.4437 q^{17} -125.054 q^{19} +182.250 q^{23} -70.0294 q^{25} -27.0000 q^{27} +156.132 q^{29} -139.632 q^{31} +31.4558 q^{33} -394.558 q^{37} -8.36039 q^{39} +197.605 q^{41} -343.294 q^{43} +66.7279 q^{45} +610.004 q^{47} -151.331 q^{51} -137.529 q^{53} -77.7401 q^{55} +375.161 q^{57} -589.436 q^{59} +247.217 q^{61} +20.6619 q^{65} +395.647 q^{67} -546.749 q^{69} -285.661 q^{71} -997.457 q^{73} +210.088 q^{75} +848.264 q^{79} +81.0000 q^{81} +210.863 q^{83} +374.000 q^{85} -468.396 q^{87} -553.487 q^{89} +418.897 q^{93} -927.176 q^{95} -903.910 q^{97} -94.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} + 12 q^{5} + 18 q^{9} - 4 q^{11} + 48 q^{13} - 36 q^{15} + 132 q^{17} - 120 q^{19} + 76 q^{23} - 174 q^{25} - 54 q^{27} - 112 q^{29} - 432 q^{31} + 12 q^{33} - 280 q^{37} - 144 q^{39} + 36 q^{41} + 128 q^{43} + 108 q^{45} + 264 q^{47} - 396 q^{51} + 268 q^{53} - 48 q^{55} + 360 q^{57} - 336 q^{59} - 504 q^{61} + 228 q^{65} + 384 q^{67} - 228 q^{69} + 396 q^{71} - 312 q^{73} + 522 q^{75} + 848 q^{79} + 162 q^{81} + 648 q^{83} + 748 q^{85} + 336 q^{87} - 612 q^{89} + 1296 q^{93} - 904 q^{95} - 2184 q^{97} - 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 12 * q^5 + 18 * q^9 - 4 * q^11 + 48 * q^13 - 36 * q^15 + 132 * q^17 - 120 * q^19 + 76 * q^23 - 174 * q^25 - 54 * q^27 - 112 * q^29 - 432 * q^31 + 12 * q^33 - 280 * q^37 - 144 * q^39 + 36 * q^41 + 128 * q^43 + 108 * q^45 + 264 * q^47 - 396 * q^51 + 268 * q^53 - 48 * q^55 + 360 * q^57 - 336 * q^59 - 504 * q^61 + 228 * q^65 + 384 * q^67 - 228 * q^69 + 396 * q^71 - 312 * q^73 + 522 * q^75 + 848 * q^79 + 162 * q^81 + 648 * q^83 + 748 * q^85 + 336 * q^87 - 612 * q^89 + 1296 * q^93 - 904 * q^95 - 2184 * q^97 - 36 * q^99

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